Improvement of the quality of a magnetoresistive (MR) element requires the increase of the MR ratio. For the purpose of increasing the MR ratio, modification of the structure of magnetoresistive elements and selection of the spacer layer materials have been carried out. For example, a magnetoresistive element including a thin spin-filter (SF) layer made of an oxide or nitride within each ferromagnetic layer or between the ferromagnetic layer and a non-magnetic spacer layer is proposed. The SF layer has a spin-filter effect of inhibiting the passage of upspin or downspin electrons, and thus increases the MR ratio. Thus, magnetoresistive elements have undergone many improvements, but are required to achieve further increase of the MR ratio.